Research Journal of Applied Sciences, Engineering and Technology 5(24): 5657-5664, 2013
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2013
Submitted: December 12, 2012
Accepted: January 21, 2013
Published: May 30, 2013
Testable Subsystems Generation for Fault Detection and Isolation Using a Structural
Matching Rank Algorithm Testability of an Electrical Circuit
Alem Saïd, Mokhtari Hicham, Ouziala Mahdi and Benazzouz Djamel
Solid Mechanics and Systems Laboratory (LMSS), University M’Hamed Bougara Boumerdes, Algeria
Abstract: In this study, an advanced way of dealing with testable subsystems and residual generation for fault
detection and isolation based on structural analysis is presented. The developed technique considers execution
issues; therefore, it has a more realistic point of view compared to classical structural approaches available in the
literature. First, theoretical aspects of structural analysis are considered and introduced. Then the way of
incorporating them to test the structural proprieties is explained. Finally, we show how the proposed (upgraded)
matching rank algorithm can be used in order to choose the most suited matching that leads to computational
sequences and detection tests. The method is demonstrated using an electrical circuit.
Keywords: Testable subsystems, Structural Analysis, Bipartite graph, Analytic Redundancy Relations ARR, Fault
Detection and Isolation FDI
INTRODUCTION
In complex systems consisting of a great number of
sensors, the obtained information starting from the
various parameters and the signals must be controlled
with an aim of the faults detection. This became a
rigorous task with the increase in the number of the
subsystems, i.e., actuators and sensors.
The structural approach constitutes a general plan
to provide information when the system becomes
complex. The main objective of the application of the
structural approach is to identify the subsystems that
present redundancy.
This study investigates the structural properties of
dynamical systems by analyzing their structural model.
The structural model of a system is an abstraction of its
behavior model in the sense that only the structure of
the constraints, i.e. the existence of links between
variables and parameters, is considered and not the
constraints themselves. The links are represented by a
bipartite graph, which is independent of the nature of
the constraints and variables (quantitative, qualitative,
equations, rules, etc.) and of the value of the
parameters. This indeed represents a qualitative, very
low level, easy to obtain, model of the system behavior.
Structural analysis is concerned with the properties
of the system’ structure model, which resorts to the
analysis of its bipartite graph. As this graph is
independent of the value of the system parameters,
structural properties are true almost everywhere in the
system parameter space.
The FDI community, Staroswiecki and Declerck
(1989), adopted the analysis of system structure,
originally developed for the decomposition of large
systems of equations for their hierarchical resolution
and structural concepts were used for the analysis of
system monitorability using complete matchings on a
graph. A recent overview of structural analysis can be
found in Blanke et al. (2006), which also provides
essential references to the field. Research on properties
of residual generators for linear differential algebraic
systems were treated in Nyberg and Frisk (2006) and
Combined structural and polynomial methods were
pursued in Krysander (2006) and Krysander and
Nyberg (2005). Techniques for active fault isolation
were treated in Niemann (2006) and suggested in a
structural context in Blanke et al. (2006). Structural
isolability properties were investigated and exemplified
in Düstegör (2005) and Düstegör et al. (2006).
In spite of their simplicity, structural models can
provide many useful information for fault diagnosis and
fault-tolerant control design, since structural analysis is
able to identify those components of the system which
are-or are not-monitorable, to provide design
approaches for analytic redundancy based residuals, to
suggest alarm filtering strategies and to identify those
components whose failure can be tolerated through
reconfiguration.
FAULT DETECTION AND
SYSTEM ANALYSIS
A typical FDI-system consists of a set of fault
detection tests associated with a fault isolation scheme
(Fig. 1). The input to the FDI-system is a set of
observations, i.e., measurements, from the monitored
Corresponding Author: Alem Saïd, Solid Mechanics and Systems Laboratory (LMSS), University M’Hamed Bougara
Boumerdes, Algeria
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Res. J. Appl. Sci. Eng. Technol., 5(24): 5657-5664, 2013
Fig. 1: A typical FDI-system
not used to obtain such matching, the set of unmatched
constraints 𝐶𝐶𝑢𝑢𝑢𝑢 ⊂ 𝐶𝐶 may be used to test the
consistency between known variables and the system’s
normal behavior. Hence, redundancy relations are
obtained from the unmatched constraints.
Solving for unknown variables in a nonlinear
system can be rather complex if done directly on the
analytical form of the constraints. Structural analysis
offers a significant shortcut. It is a method to determine
possible ways to solve a set of constraints without
actually doing so. Making a graph representation of the
relations between constraints and unknown variables
makes it possible to seek through a graph to determine
how one could solve for unknown variables. The result
of structural analysis is a receipt that, in a symbolic
form, describes how unknown variables could be
calculated from known variables, using the system
constraints. Analytical expressions are not used until a
complete structural solution is found. This dramatically
reduces the complexity of finding parity equations for
fault diagnosis.
The salient feature of the structural analysis
approach is that graph theory exists which can be
employed to find all possible ways the set of system
constraints can be matched to unknown variables
(Dulmage and Mendelsohn, 1959). As sets of
unmatched constraints, in general, differ from matching
to matching, structural analysis can determine the entire
STRUCTURAL MODEL
set of possible parity relations.
Being very useful as a first step of the analysis, the
This section provides a brief overview of the main
results of structural analysis are, however, only
concepts in structural analysis. It provides a baseline for
indicative of the existence of the associated analytical
further extensions to enhanced structural faults
results. The existence of a structural parity relation does
detection and isolation.
not guarantee the existence of an analytic counterpart.
The essential idea in analytic fault diagnosis is to
establish relations to test whether measured and other
Non-existence in the structural domain does, however,
known variables satisfy all relations that describe the
imply non-existence also in the analytical domain.
system’s normal behavior. If this is not the case, some
Structural concepts were studied early in the
violation of the normal behaviour has occurred, i.e., one
applied mathematics community and various theoretical
or more faults are present in the system. Relations that
algorithms were developed in Hopcroft and Karp
can be used for such testing are referred to as
(1973) and Izadi-Zamanabadi et al. (2003). Structural
redundancy relations.
analysis was and is used intensively in chemical
Let a system be described by a set X of unknown
engineering for solving large sets of equations (Leitold
variables, a set K of known variables and a set of
and Hangos, 2001; Unger et al., 1995). The structural
constraints C on these variables.
approach
and the features it offers for analyzing
Then there may exist a set 𝐶𝐶𝑚𝑚 ⊆ 𝐶𝐶 from which all
monitoring and diagnosis problems were first
variables in X can be determined. A system that has this
introduced in Staroswiecki and Declerck (1989).
property is said to have a complete matching on the
Extensions to the analysis of reconfigure ability and
unknown variables. If any constraints exist that were
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system and the output is a diagnosis statement. The
diagnosis statement contains a collection of faults that
can be used to explain the observations.
Given a set of observations, y, the outcome of a
detection test τ i is a binary fault detection result, d i ,
equal to, for instance,1 if the test has alarmed, or equal
to 0, otherwise.
To enable fault isolation, different detection tests
typically monitors different faults and thus different
parts of the system. Each fault detection test typically
utilizes a subset of the observations in order to
determine if any fault is present in its monitored part of
the system.
Common traditional approaches for construction of
fault detection tests are for example limit checking, i.e.,
to check if a sensor is within its normal operating range,
or to employ hardware redundancy. For instance, if two
sensors are used to measure the same physical quantity,
it is possible to test if one of the sensors is faulty by
comparing the values of the sensors. Another approach,
providing potentially increased diagnosis performance
and in which the need of additional, redundant,
hardware is avoided, is to use detection tests based on
residuals. Detection tests based on residuals will be
generated further in the next section using a structural
approach.
Res. J. Appl. Sci. Eng. Technol., 5(24): 5657-5664, 2013
fault-tolerance emerged in Izadi-Zamanabadi and
Staroswiecki (2000) and Staroswiecki and Gehin
(2000).
The structural analysis approach was presented in a
digested form in Düstegör et al. (2004). Structural
analysis has hence evolved during several decades.
However, the salient features of the theory and the
possibilities it offers have only become apparent to a
larger community in the field of automation and
automatic control over the last few years (Åström et al.,
(2001; Izadi-Zamanabadi et al., 2003) with applications
reported in, e.g., Izadi-Zamanabadi and Staroswiecki
(2000).
Structure as a bi-bipartite graph : The model
structural represents the links between a set of variables
and a set of constraints. It is an abstraction of the
behavior model because it describes simply which
variables are connected to which constraints. Hence, the
structural model presents the basic features and
properties of a system which are independent of its
parameters. The behavior model of a system is defined
by a pair (C, Z) where Z is a set of variables and
parameters and C is a set of constraints (Blanke
et al., 2006; Düstegör et al., 2004).
This structure can be represented by a graph
bipartite. A graph is bipartite if its vertices can be
separated in two disjoin sets C and Z in such a way that
every edge has one endpoint in C and the other one in
Z.
To note that the graph bipartite is undirected graph,
which can be interpreted as follows: All the variables
and parameters connected with a given constraint vertex
has to satisfy the equation or rule this vertex represents.
This graph allows representing the structure of rather
general models including both differential and algebraic
constraints.
In the bipartite graphs the vertices of Z are
represented by circles while constrained the vertices of
C will be represented by bars.
Example 1: For a better comprehension, the example
above will be used, (Ploix et al., 2005) (Fig. 2).
Fig. 2: Electrical circuit
Fig. 3: Structure graph of the electrical system
Table 1: Incidence matrix of the system
Unknown variables
-------------------------Variables/
constraints
R
C
s
𝑉𝑉1̇
C1
1
C2
C3
1
1
C4
C5
1
C6
1
Known variables
---------------------------------------------e
v1
v2
i1
i2
i3
1
1
1
1
1
1
1
1
1
1
1
To consider an electrical system where the courant
and the tension are the variables propagating in the
circuit.
The system consists of the components {Resistor
R, capacitor C, Output charge s, Generator e, Sensors i,
v). A continuous model with variable in continuous
time is given by the following constraints:
•
•
•
•
•
•
Resistor C 1 : 𝑒𝑒 − 𝑣𝑣1 = 𝑅𝑅𝑖𝑖1
Node rule C 2 : 𝑖𝑖1 = 𝑖𝑖2 + 𝑖𝑖3
𝒅𝒅
Capacitor C 3 : 𝑖𝑖2 = 𝐶𝐶 𝑣𝑣1
𝒅𝒅𝒅𝒅
Tensions rule C 4 : 𝑣𝑣1 = 𝑣𝑣2
Charge C 5 : 𝑣𝑣1 = 𝑠𝑠
Derivative constraint C 6 : 𝑉𝑉1̇ =
𝑑𝑑
𝑑𝑑𝑑𝑑
𝑣𝑣1
The graph bipartite of our system is represented in
the Fig. 3 as follows:
We can also represent our system in the form of
incidence matrix, Table 1. Every column of the matrix
correspond to a circle-vertex and every row correspond
to a bar-vertex, the variables that are connected to a
constraint is marked by 1 in its box as is given in the
following:
Known and unknown variables: System variables and
parameters can be decomposed into known (Internal)
and unknown (external). System inputs and outputs are
examples of variables that are usually known. In the
same way, parameters that are entries for the model or
were previously identified are certainly known. Thus,
known variables are available in real-time and they can
directly be employed in the faults diagnosis.
Unknown variables are not directly measured,
although there might exist some way to compute their
value from the values of known ones (calculability). In
the electrical circuit example, the last five columns of
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Res. J. Appl. Sci. Eng. Technol., 5(24): 5657-5664, 2013
the incidence matrix {e, v 1 , v 2 , i 1 , i 2 , i 3 } correspond to
known variables, while the first four ones correspond to
unknown variables {R,C, s,𝑉𝑉1̇ }, (the components and
the derivative state).
Variable set is portioned into:
𝑍𝑍 = 𝑘𝑘 𝑈𝑈 𝑋𝑋
Or: K (known variable) and X (unknown variable)
And: 𝐶𝐶 = 𝐶𝐶𝐾𝐾 𝑈𝑈 𝐶𝐶𝑋𝑋
C K constraint, which links only the known
variables and C X includes constraints in which at least
one unknown variables appears.
MATCHING ON A BI-PARTITE GRAPH
The basic tool for the structural analysis is the
concept of matching founded on a bipartite graph. In
other words, a matching is a causal assignment, which
associates unknown variables of the system with the
constraint of system from which they can be calculated.
The unknown variables that cannot be matched cannot
be calculated. The variables, which can be matched
several ways, can be determined by various ways,
which provide means for fault detection and a
possibility for reconfiguration (Düstegör et al., 2004).
On a Bipartite graph, the matching is a set of
arcs/edges such that any two edges have no common
node (neither in C nor in Z) and it is represented by an
arc in Gras. This is given as follows, in Fig. 4, an
example of two possible matchings for the electrical
circuit system:
In the incidence matrix, the matching is
represented by one ①. Only one ① is authorized in
same column and line.
Matching algorithm: We can count many algorithms
to find the maximal matching, like the classical
maximal matching algorithm, the maximal flow
matching, Hungarian method (Dulmage and
Mendelsohn, 1963; Hopcroft and Karp, 1973).
However, we will use an efficient and upgraded
algorithm based on ranks to find the matchings
considering his effectiveness and his simplicity of
application on a system. See for basics, Blanke et al.
(2006). In our ameliorated version of the algorithm, it is
possible to generate all non-matched constraints in each
iteration, this is useful for diagnosis and for tests
generation.
Ranking algorithm
Given: an incidence matrix or a structural graph
•
•
•
•
•
•
Fig. 4: Possible matching on the bipartite graph of the electrical system
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Mark all know variables
i=0
Fine all constrain in the current table with exactly
one unmarked variable. Associate them the rank i
and mark these constraints as well as the
corresponding variable.
Set i = i + 1
If there are unmarked constraints where all
variables with rank i, mark them and link them to
the pseudo-variable ZERO
If there are unmarked variables or constraints,
continue with step2
Res. J. Appl. Sci. Eng. Technol., 5(24): 5657-5664, 2013
The constraint propagation algorithm applied to the
electrical circuit system is exhaustively given as
follows:
Since, {e, v 1 , v 2 , i 1 , i 2 and i 3 } are known, only the
variable set {R,𝑉𝑉1̇ , C, s} has to be matched.
•
•
•
•
Starting set (rank 0): {e, v 1 , v 2 , i 1 , i 2 , i 3 }
First step (rank 1): match 𝑉𝑉̇1 with C 6 and match s
with C 5 .
Second step (rank 2): match C with C 3 and match
R with C 1 .
It is clear that all our unknown variables are matched.
System’ canonical decomposition: The graph theory
recalls a classical result from bipartite graph, (Dulmage
and Mendelsohn, 1959), which states that any finitedimensional graph can be decomposed into three subgraphs with specific properties, respectively associated
with an over-constrained, a just-constrained and an
under-constrained subsystem. This decomposition is
canonical, i.e., for a given system, it is unique. The
three subsystems play a major role in the analysis of the
system
structural
properties:
Observability,
Controllability and Monitorability.
A graph (C, Z) is called:
•
•
•
Over-constrained: if there is a complete matching
on the variables Z but not on the constraint C,
Just-constrained: if there is a complete matching
on the variables Z and on the constraints C,
Under-constrained: if there is a complete
matching on the constraints C but not on the
variables Z.
PROPERTIES OF STRUCTURAL ANALYSIS
In addition, this is feasible through designing fault
detection and isolation algorithms based on the
analytical redundancy relation.
Analytical redundancy ARR based fault diagnosis
tries to identify faults by comparing the actual behavior
of the system, with the theoretical behavior described
by the system constraints. They are given in the means
of only known variables. ARRs are the constraints that
express this redundancy. This redundancy is
represented in the following with the over-constrained
and unmatched constraints that denote the testable
subsystems.
The ARR should have the following properties:
•
•
•
Robust: i.e., insensitive to unknown input and
unknown parameters.
Sensitive to faults: this insures that they are not
satisfied when faults are present, so that there is no
missed detection.
Structured: this insures that in the presence of a
given fault, only subsets of the ARR are not
satisfied, thus allowing recognizing, the fault
which occurred.
The following two necessary conditions meant for
which a fault to be monitorable are equivalent:
•
•
X is structurally observable in the system
It belong to the over-constrained part of the system
(C, Z), i.e., that the unknown variables must be
observable (adding sensors) and it belongs to an
overstrained subset (complete matching on the
variables Z).
Design
of
analytic
redundancy
relations:
Redundancy relations are sub-graphs of the structure
graph, which are associated with complete causal
matching of the unknown variables and associated with
the over-constrained subsystem of a reduced bi-partite
graph. Redundancy relations are composed of alternated
• All the unknown variables are reachable from the
chains, which start with known variables and end with
known ones
non-matched constraints, here the chosen output is
• The over-constrained and the just-constraint
labeled ZERO.
subsystems are causal
Designing a set of residuals requests for building
• The under-constrained subsystem is empty
maximal matching on the given structural graph, under
derivative causality and identifying the redundancy
To simplify, to ensure that the system is
relations as the non-matched constraints in which all the
observable, it is necessary that, it be over-constrained or
unknowns have been matched. The algorithm to find
just-constrained. Moreover, it is necessary that all the
maximal matching has been previously presented.
unknown variables are reachable (Blanke and
In our studied example, the incidence matrix
Staroswiecki, 2006), i.e., can be calculated by means of
obtained
after applying the rank algorithm is in the
the known variables.
form below, Table 2:
It is clear that constraints C 2 and C 4 are the nonMonitorability: A system is said to be monitorable, if
matched constraints, where all the variables are
it can be determined using only known variables,
whether the system constraints are satisfied or not.
matched.
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Observability: Observability informs us about the
unknown variables of our system that can be checked
and observed.
Structural Observability condition (Blanke et al., 2006):
Res. J. Appl. Sci. Eng. Technol., 5(24): 5657-5664, 2013
Table 2: Incidence matrix after matching
Internal variables External variables
---------------------- ---------------------------------------Variables/
constraints R
C 𝑉𝑉1̇
s e
v1
v2
i1
i2 i3
C1
1
1
1
①
C2
1
1 1
C3
1
① 1
C4
1
1
C5
1
①
C6
1
①
Rank
2
Zero
1
Zero
0
0
Constraint C 2 and C 4 are used for the design of
redundancy relations by replacing all the matched
variables by the known variables, the results are as
following:
The given calculation order of unknown variables
from knows is as follow:
C4: ν 1 = ν 2 → C5: ν 1 = s → C2: i 1 + i 2 + i 3 → C6:
𝑑𝑑
𝑑𝑑
𝑉𝑉̇ = 𝑣𝑣1 → C3: 𝑖𝑖2 = 𝐶𝐶 𝑣𝑣1 → C1: 𝑒𝑒 − 𝑣𝑣1 =
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
𝑅𝑅 𝑖𝑖1→ Zero
By replacing the known variables we have:
C 1 :0 = 𝐶𝐶𝑣𝑣1 + 𝑅𝑅 𝑖𝑖1 ‒ 𝑒𝑒 = 𝑟𝑟(𝑡𝑡)
where, r(t): is a residual obtained by the analytical
redundancy relation, the residual is an evaluation of
the ARR, it is equal to 0 if the system is free-fault
and to a non-zero value in the presence of a fault
(fault detection). It is s-called also, the fault indicator
or fault signature, if two faults have the same fault
signature, they are not structurally distinguishable
(fault isolation).
We can note that Kirchhoff laws are generally
the redundant constraints in circuit analysis and they
are commonly used to calculate residues.
The order of operations on constraints is(coming
from the last) C 1 (C 3 (C 6 (v 1 )),C 2 (i 2, i 3 ),C 3 (C 4
(v 2 ))) → zero
We remark that v 2 and v 1 are redundant sensors
and it is similar for i 1 and (i 2 ,i 3 ). Hence, it is well
known that hardware redundancy is generally the
commonly used approach in circuit tests, the next
section deals with redundancy elimination in testable
subsystems generated using a structural methodology.
function defined on a space of known variables, based
on all the analytical relations of the elementary models
belonging to the subsets.
In the following, a short survey on related works is
given. Armengol et al. (2009) resumes all the methods
used in the literature to generate the testable subsystems
in the context of structural analysis.
Krysander et al. (2008) proposed an algorithm for
finding all MSOs (minimal structurally overconstrained subsystems), i.e., testable subsystems TSS.
This algorithm is based on the Dulmage-Mendelsohn
decomposition, (Dulmage and Mendelsohn, 1959) and
on a top-down approach. It starts with the entire model
and then reduces the size of the model step by step until
a TSS remains.
A handicap of this method is that, the same TSS
can be found more than once. Therefore, this algorithm
is not optimal in terms of efficiency. Krysander et al.
(2008) presented improvements in order to increase the
efficiency.
Even if the complexity of this method is lower than
for the last approach, the main disadvantage of this
method is that it cannot take into account the notion of
deductibility/reachability of variables, Blanke and
Staroswiecki (2006): all variables of the system are
considered as deductibles and, therefore, some ARRs
may not be achievable. Of course, systems with
branchings cannot be managed.
In this study, we combine the rank algorithm with
the approach proposed in Krysander et al. (2008) to
ameliorate TSS generation. The main idea is using the
elimination of the redundancy in the incidence matrix
and reduces the number of constraints used to the
minimal; this will generate the basic testable
subsystems defined in Ploix et al. (2005).
Taking in account the following definition of
deductibility (calculability):
Definition 1: Two constraints C1 and C2 defined on a
set of physical variables V will be considered as
equivalent𝜎𝜎(𝐶𝐶1) = 𝜎𝜎 (𝐶𝐶2). It is denoted C1 ⇔ C2.
The set of references 𝜎𝜎(𝐶𝐶), also called constraint
support, containing either the name of the constraint if
it is related to only one component state, or, elsewhere,
the names of all the constraints that compose C.
GENERATION OF TESTABLE SUBSYSTEMS
A testable subsystem is a set of constraints, which
leads to an Analytical Redundancy Relation (ARR). An
ARR is a static or dynamic time evolution constraint
that contains only known variables and it is equal to
zero (0) when the system operates according to its
normal operation model.
Any given subsystem model cannot always be
checked because testing the consistency of a subset of
elementary models requires that some physical
variables to be known. A test, also called a detection
test, dedicated to a subsystem model is a Boolean
Definition 2: A constraint C2 overestimates
(miscalculates) a constraint C1 if 𝜎𝜎 (𝐶𝐶1) ⊂ 𝜎𝜎 (𝐶𝐶2), It
is denoted 𝐶𝐶1 ⊂ 𝐶𝐶2.
Constraints overestimating others and equivalent
constraints have to be removed because they are not
minimal. This is the main idea in Krysander et al.
(2008) and Yassine et al. (2008), they called it
redundant constraints.
Recall the example 1, the incidence matrix for the
reduced model of electrical circuit (without
miscalculated and redundant constraints and variables)
is as follow, Table 3:
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Res. J. Appl. Sci. Eng. Technol., 5(24): 5657-5664, 2013
Table 3: Incidence matrix after re-arrangement and reduction
Internal variables
External variables
----------------------------------------------------Variables/
R
C
s
e
v1
i1
i2
constraints
𝑉𝑉1̇
\
1
1
1
①
C3
1
1
①
C6
1
①
C5
1
①
Ploix et al. (2005). The paper gives a short
investigation on related works and commonly utilized
terminology on testable subsystems generation using a
structural approach.
In future works, one can include some causal
interpretations on computational sequences, i.e.,
derivative and algebraic loops. Amelioration on the
proposed algorithm o fit with large-scale systems with
dynamic behaviors will be also considered. It is
possible to take into account the exponential
complexity of the iterative algorithm and proposing a
heuristic approach to deal with an optimization on the
combination of all the obtained basic matchings in each
iteration.
REFERENCES
Fig. 5: Bipartite graph of reduced system with a maximal
matching on variables
One finds that, by establishing all the possible
matchings, it is possible to generate all the detection
tests (basic tests), which will be used for the design of
all Testable subsystems.
Figure 5, illustrates a maximal matching on the
reduced model.
It is proposed that: by tracking all the linked edges
on the graph, starting from a matched variable and
passing by a constraint, allows us to tracing the order of
calculability directly from the Bipartite graph and
generate a testable subset (a basic subset),this is called a
mapping. As a result, by establishing all the possible
mappings on Fig. 5, one can get all the testable
subsystems (the sets of all possible basic tests in
different calculation orders) i.e., the over-determined
subsets of constraints, which can be used directly in the
FDI-scheme defined in Fig. 1.
CONCLUSION
The structural analysis enables the exploration of
local redundancy of the system, it also allows
highlighting the possibilities for monitoring through the
structural properties such as, Monitorability and
determination of calculation sequences, the result is a
residue and allows generating all the testable
subsystems. Analysis of residues structure has in
purpose to assess the faults detectability and isolability,
but it can also suggest what sensors should be
implemented in order to change the status of the system
components, to move a non-observable one to an
observable (Alem and Benazzouz, 2013).
An ameliorated ranking algorithm is used to
generate the testable subsets and produce detection tests
by eliminating the miscalculated and equivalent
(redundant) constraints in the structured model. The
efficiency of the proposed method is confirmed along
the paper using an electrical circuit system. The
obtained results are clearly compared with those in
Alem, S. and D. Benazzouz, 2013. Optimal sensor
placement for fault detection and isolation by the
structural adjacency matrix. Int. J. Phys. Sci., 8(6):
225-230.
Armengol, J., A. Bregon, T. Escobet, E. Gelso, M.
Krysander, M. Nyberg, X. Olive, B. Pulido and L.
Trave-Massuyes, 2009. Minimal structurally overdetermined sets for residual generation: A
comparison of alternative approaches. Proceedings
of the 7th IFAC-Safeprocess 09. Barcelona, Spain,
pp: 1480-1485.
Åström, K., J. Albertos, P. Blanke and M. Isidori, 2001.
A
Control
of
Complex
Systems.
In:
Schaufelberger, W. and R. Sanz (Eds.), Springer,
London/New York.
Blanke, M. and M. Staroswiecki, 2006. Structural
design of systems with safe behavior under single
and multiple faults. Proceedings of the 16th IFAC
Symposium Safeprocess. Beijing, China, pp: 511516.
Blanke, M., M. Kinnaert, J. Lunze and M. Staroswiecki,
2006. Diagnosis and Fault-Tolerant Control. 2nd
Edn., Springer, Berlin, Heidelberg.
Dulmage, A.L. and N.S. Mendelsohn, 1959. A structure
theory of bipartite graphs of finite exterior
dimension. Trans. Royal Soc. Canada Ser., 53:
1-13.
Dulmage, A.L. and N.S. Mendelsohn, 1963. Two
algorithms for bipartite graphs. J. Soc. Indus. Appl.
Math, 11(1): 183-194.
Düstegör, D., 2005. Structural analysis for FDI:
Algorithmic issues. Ph.D. Thesis, University of
Science and Technology of Lille (French), France.
Düstegör, D., E. Frisk, V. Cocquempot, M. Krysander
and M. Staroswiecki, 2006. Structural analysis of
fault isolability in the DAMADICS benchmark.
Control Eng. Pract., 14(6): 597-608.
Düstegör, D., V. Cocquempot and M. Staroswiecki,
2004. Structural analysis for fault detection and
identification: An algorithmic study. Proceedings
of the 2nd Symposium on System Structure and
Control (SSSC’04). Oaxaca, Mexico.
5663
Res. J. Appl. Sci. Eng. Technol., 5(24): 5657-5664, 2013
Hopcroft, J.E. and R.M. Karp, 1973. An n/sup 5/2
algorithm for maximal matchings in bipartite
graphs. SIAM J. Comput., 2(4): 225-231.
Izadi-Zamanabadi, R. and M. Staroswiecki, 2000. A
structural analysis method formulation for faulttolerant control system design. Proceedings of the
39th IEEE Conference on Decision and Control.
Sydney, Australia, pp: 4901-4902.
Izadi-Zamanabadi, R., M. Blanke and S. Katebi, 2003.
Cheap diagnosis using structural modeling and
fuzzy-logic based detection. Control Eng. Practice,
11(4): 415-422.
Krysander, M., 2006. Design and analysis of diagnosis
systems using structural methods. Ph.D. Thesis,
Linköping University, Sweden.
Krysander, M. and M. Nyberg, 2005. Fault isolability
prediction of diagnostic models. Proceedings of
16th International Workshop on Principles of
Diagnosis DX-05. Pacific Grove, CA, USA.
Krysander, M., J. Aslund and M. Nyberg, 2008. An
efficient algorithm for finding minimal overconstrained subsystems for model-based-diagnosis.
IEEE T. Syst. Man Cy. A, 38(1): 197-206.
Leitold, A. and K.M. Hangos, 2001. Structural
solvability analysis of dynamic process models.
Comput. Chem. Eng., 25(11-12): 1633-1646.
Niemann, H.H., 2006. A setup for active fault
diagnosis. IEEE T. Automat. Contr., 51(9):
1572-1578.
Nyberg, M. and E. Frisk, 2006. Residual generation for
fault diagnosis of systems described by linear
differential-algebraic equations. IEEE T. Automat.
Contr., 51(12): 1995-2000.
Ploix, S., M. Desinde and S. Touaf, 2005. Automatic
design of detection tests in complex dynamic
systems. Proceeding of 16th IFAC World
Congress. Prague, Czech Republic, July 4-8.
Staroswiecki, M. and P. Declerck, 1989. Analytical
redundancy in nonlinear interconnected systems by
means of structural analysis. Proceedings of the
IFAC Symposium on Advanced Information
Processing in Automatic Control (AIPAC’89).
Nancy, France, pp: 23-27.
Staroswiecki, M. and A.L. Gehin, 2000. From control
to supervision. Proceedings of the IFAC
Symposium Safeprocess’00. Budapest, Hungary,
Vol. 1.
Unger, J., A. Kröner and W. Marquardt, 1995.
Structural analysis of differential-algebraic
equation systems-theory and applications. Comput.
Chem. Eng., 19(8): 867-882.
Yassine, A., S. Ploix and J.M. Flaus, 2008. A method
for sensor placements taking into account
diagnosability criteria. Appl. Math. Comput. Sci.,
18(4).
5664